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The Florida senator sits on the Senate Foreign Relations Committee

Meanwhile, Trump critics and allies raced to define the international business executive

Washington (CNN)Opposition to Rex Tillerson as the next secretary of state heightened Tuesday as Marco Rubio, a key Republican vote on the panel that oversees his confirmation, said he had "serious concerns about his nomination."

Donald Trump said Tuesday morning he would nominate the ExxonMobil CEO to the nation's top diplomatic post, and the announcement did not split Washington along strictly partisan lines. Some Republican foreign policy hawks, like Rubio, have expressed discomfort with Tillerson's ties to Russia.

Rubio sits on the Senate Foreign Relations Committee, the panel that will have to approve the nominee before the full Senate can vote on Tillerson. Republicans will likely only hold a one-seat advantage on the committee, meaning that Democrats along with any single Republicans can possibly scuttle the nomination.

"While Rex Tillerson is a respected businessman, I have serious concerns about his nomination," Rubio said in a statement. "The next secretary of state must be someone who views the world with moral clarity, is free of potential conflicts of interest, has a clear sense of America's interests, and will be a forceful advocate for America's foreign policy goals to the president, within the administration, and on the world stage. I look forward to learning more about his record and his views.‎"

Meanwhile, Trump critics and allies raced to define the international business executive. Sen. Amy Klobuchar, a Democrat from Minnesota, said the Republicans' initial reaction to Tillerson indicated there was "bipartisan concern" about his Russia ties.

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"I'm sure he has excellent business credential, but Donald Trump has chosen to put someone in as the nominee who does not have foreign policy or diplomatic experience and has these ties to Russia," she told CNN's Poppy Harlow on "New Day."

And Rudy Giuliani, the close Trump associate who was passed over for the secretary of state posting, told CNN's Chris Cuomo that while he did not know Tillerson well, he was confident in him and would not say that he would be a better State candidate than Tillerson.

"It's hard to say: Who would've been better as secretary of state? You find that out three or four years from now," he said.